Introduction to the Theory of Dirichlet Forms

Abstract

A Dirichlet form is a generalization of the energy form \(f\mapsto \int _\Omega |\nabla f|^2 d\lambda \) introduced in the 1840s especially by William Thomson (Lord Kelvin) (cf. Temple, 100 Years of Mathematics, 1981, [351], Chap. 15) in order to solve by minimization the problem without second member \(\Delta f=0\) in the open set \(\Omega \) (Dirichlet principle). Riemann adopted the expression Dirichlet form (Riemann Grundlagen fur eine allgemeine Theorie der Funktionen einer veranderlischen komplexen Grosse, 1851, [314]). The generalization now known as a Dirichlet form keeps the notion in the same relationship with the semigroup as the energy form holds with the heat semigroup.